A102887 Decimal expansion of Integral_{x=0..1} log(gamma(x))^2 dx.

1, 8, 6, 6, 3, 1, 7, 0, 8, 3, 7, 9, 3, 5, 6, 2, 0, 8, 0, 9, 9, 2, 9, 6, 7, 9, 3, 7, 9, 7, 8, 2, 8, 9, 7, 3, 9, 8, 0, 0, 4, 0, 4, 1, 8, 6, 7, 9, 5, 3, 3, 8, 8, 0, 9, 4, 0, 5, 5, 1, 4, 4, 9, 5, 9, 3, 0, 4, 0, 9, 6, 5, 9, 8, 4, 9, 0, 5, 6, 3, 0, 3, 4, 7, 5, 5, 2, 3, 9, 8, 6, 0, 2, 9, 2, 5, 7, 2, 5, 0, 8, 5

Offset: 1

Author

Eric W. Weisstein, Jan 15 2005

Keywords

• nonn
• cons

Comments

Also equals (1/6)*log(2*Pi)^2 + 2*log(A)*log(2*Pi) - (1/6)*gamma*log(2*Pi) + Pi^2/48 + 2*gamma*log(A) + zeta''(2)/(2*Pi^2) (with A the Glaisher-Kinkelin constant). - Jean-François Alcover, Apr 29 2013

Examples

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References

• George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 236.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• M. L. Glasser, A note on Beukers's and related integrals, Amer. Math. Monthly 126(4) (2019), 361-363.

Crossrefs

Cf. A001620, A074962, A075700, A201994.

Programs

Mathematica

EulerGamma^2/12 + Pi^2/48 + (EulerGamma*Log[2*Pi])/6 + Log[2*Pi]^2/3 - ((EulerGamma + Log[2*Pi])*Zeta'[2])/Pi^2 + Zeta''[2]/(2*Pi^2)

PARI

intnum(x=0,1, log(gamma(x))^2) \\ Michel Marcus, Aug 27 2015

Formula

Equals gamma^2/12 + Pi^2/48 + (gamma*log(2*Pi))/6 + log(2*Pi)^2/3 - ((gamma + log(2*Pi))*zeta'(2))/Pi^2 + zeta''(2)/(2*Pi^2).

Equals -Integral_{x=0..1, y=0..1} log(gamma(x*y))^2/log(x*y) dx dy. (Apply Theorem 1 or Theorem 2 in Glasser (2019).) - Petros Hadjicostas, Jun 30 2020